The science of fluxions was Sir Isaac Newton's terminology for the new field of science known today as calculus. Newton and German mathematician Leibnitz appear to have discovered the principles of calculus in about the same time; but Leibnitz published his work first. For years, there was friction between the two countries, England and Germany regarding which country was to take credit for the discovery of calculus. In the final analysis, it appears that both men arrived at their findings at about the same time and independently of each other. Both men appear to have learned from Egyptian, Indian and ancient Greece sources.
The name calculus is derived from the Latin or Roman term meaning pebbles which were a type of counting stone.

The term fluxion was Sir Isaac Newton's term for the science of calculus. His book "Method of Fluxions was published in posthumously in 1736, although it was completed much earlier in 1671. Whether Leibnitz's or Newton's authorship of the science is accepted as being primary, the fact remains that the subject of calculus is the most powerful mathematical invention of modern times.
There are two major types of calculus, known as Infinitesimal Calculus and that part of the total which is called Differential Calculus. Both types are built on a foundation on analytic geometry and are related by the Fundamental Theorem of Calculus.

In simple terms, the theorem states that the sum of infinitesimal changes over time or some other quantity will add up to the net change.
To use a living plant as an example, as it grows, you can see the difference in size, or the increase over a period of several days and you could measure the growth, using conventional means. However, if you were to measure the difference in size after only ten seconds, it would be much more difficult to determine. If you then were to attempt to determine not the amount of growth over the time period, but the rate of growth over those seconds, you would not be able to do that using algebraic terms.
The science of calculus allows you to determine the rate of change for infinitesimally small amounts. There are a number of functions which impact the rate of change for the above example of a growing plant; things such as amount of sunlight, water, the temperature and others. However, if all other things remain equal, the variable for the rate of growth is time. By using calculus these variables can be determined. Most of the work in calculus is done by graphing formulas in order to determine the slope of the rate of change.
Calculus can be used in any field of science where a mathematical model can be designed and an optimal solution is desired.

This includes physical sciences, computer sciences, business, medicine, engineering, economics and statistics. Although in the past, calculus has been considered too complicated to study as an individual subject without the benefit of a teacher's direction, internet courses are readily available and can be studied as a self-paced course of study.